It seems to me that this should be a fairly easy bug to fix. Could someone please direct me to where I could file a bug report?
In the mean time, I did come up with an inelegant workaround. I had clicked the "Try it out" button for using the new control panel on my Chrome browser. I'd tried clearing cookies for the nightbot domain but that did not help.
I did find that I was able to get to the other streamer's song_requests page with Firefox without being redirected. This is a bit clunky as I don't have my bookmarks synchronized between Chrome and Firefox and I've organized all of the links for tracks I play in Chrome.
Obviously, I'd love for this behavior to get fixed but I expect that a forum post isn't going to make it happen. I'll need to get the bug properly reported and acknowledged to get things in motion.